# zpk

Convert unconstrained MPC controller to zero/pole/gain form

## Description

Use the Model Predictive Control Toolbox™ zpk function to convert an unconstrained MPC controller with defined sample time to zero/pole/gain form (see mpc for background). The returned controller is equivalent to the original MPC controller mpcobj when no constraints are active. You can then use Control System Toolbox™ software for sensitivity analysis and other diagnostic calculations.

To create or convert a generic LTI dynamical system to zero/pole/gain form instead, see zpk and Dynamic System Models.

example

kzpk = zpk(mpcobj) returns the linear discrete-time dynamic controller kzpk, in zero/pole/gain form. kzpk is equivalent to the MPC controller mpcobj when no constraint is active.

## Examples

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Create a plant, a corresponding MPC object, and convert it to zero/pole/gain form.

mpcverbosity off;          % turn off mpc messaging
plant=tf(1,[1 1],0.2);     % create plant (0.2 seconds sampling time)
mpcobj=mpc(plant,0.2);     % create mpc object (0.2 second sampling time)

kzpk=zpk(mpcobj)           % convert to zpk form show the controller's poles and zeroes

kzpk =

From input "MO1" to output "MV1":
0.45205 z^2 (z-1.5)
-----------------------------
(z-1) (z-0.02575) (z+0.02485)

Sample time: 0.2 seconds
Discrete-time zero/pole/gain model.

The poles are all inside the unit circle, except the one in z=1. The position of this pole , which is due to the fact that the default noise model is an integrator, causes the controller static gain to approach infinity, in turn allowing near perfect tracking of the output reference signal.

## Input Arguments

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Model predictive controller, specified as an MPC controller object. To create an MPC controller, use mpc.

## Output Arguments

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Zero/pole/gain form of the MPC controller mpcobj when no constraint is active. This is also equivalent to zpk(ss(mpcobj))

Note

If mpcobj is set to use custom state estimation, then tf returns a static gain feedback matrix from the states of the augmented discrete-time plant, including previous values of the manipulated variables, to the manipulated variables.

## Version History

Introduced before R2006a